Fatigue crack propagation threshold values have been determined with two experimental methods, it., the constant R method and the constant K, , method. Three materials, namely A17075-T735 1 and Ti6Al4V STA in the LT-and TL-orientations, and a Ti-turbine disk material (IMI 685) in the CR-orientation, were investigated.The paper is divided into 3 parts. In the first part the test conditions, the experimental results and the conclusions drawn from the experimental results are presented, namely that the three different functional dependencies of AK* on R cannot be reconciled with present continuum mechanics concepts. In the second part, some facts used in conjunction with the da/dN -AK,, methodology are applied to the nonpropagation condition AKth. Parameters such as KO,, the threshold AKT, and a parameter "K,; are investigated by numerical modelling of their individual in3uence on the AKth versus R curves. This modelling work shows that the individual AKth versus R curves are primarily dependent on the KO, behavior of the respective material. Further, it is shown that the threshold AKT is a constant value, independent of any particular cyclic loading condition. In the third part of the paper, the AK,, concept is applied to the experimental results obtained in the first part. Using either experimentally or semiempirically determined KO, functions and the measured AKr values, the A K~ versus R curves of the three materials investigated were accurately reconstructed. It follows that the AKth versus R curves of the individual materials are the natural consequence of the driving force for fatigue crack propagation, namely AKem NOMENCLATURE FCP = fatigue crack propagation F-, F, , , = minimum and maximum load applied during a cycle K,,, K,,, = minimum and maximum stress intensity factor of a cycle AK = Kmx -Kmin = stress intensity range AKth = stress intensity range when FCP just ceases AKr = materal property describing the maximum AK which causes no continuous FCP KO, = the opening instant of the stress intensity factor under arbitrary cyclic loading K,,(max. value) = upper bound value of KO, at a given K,,, AKe, = effective part of AK front during a cycle F,,, K,, = minimum load and minimum stress intensity factor experienced by the crack KZaX = lowest K, , at which FCP occurs under zero-tension cycling R = Kmin/Kmax = stress intensity factor ratio R, = R-ratio at which Kmin = K,, and AK = AKT 4, ni = coefficients of polynomial equations 'Y(R), @ (Kmax) = polynomial functions